A Best Proximity Point Result in Modular Spaces with the Fatou Property

Abstract

Consider a nonself-mapping 𝑇∶𝐴→𝐵, where (𝐴,𝐵) is a pair of nonempty subsets of a modular space 𝑋_𝜌. A best proximity point of 𝑇 is a point 𝑧∈𝐴 satisfying the condition: 𝜌(𝑧−𝑇𝑧)=inf{𝜌(𝑥−𝑦)∶(𝑥,𝑦)∈𝐴×𝐵}. In this paper, we introduce the class of proximal quasicontraction nonself-mappings in modular spaces with the Fatou property. For such mappings, we provide sufficient conditions assuring the existence and uniqueness of best proximity points.

1. Introduction and Preliminaries

Through this paper, we denote by ℕ the set of positive integers including zero. Let 𝑋 be a vector space over ℝ. We denote by 0_𝑋 its zero vector. According to Orlicz [1], a functional 𝜌∶𝑋→[0,∞] is said to be modular, if, for any pair (𝑥,𝑦)∈𝑋², the following conditions are satisfied:(i)𝜌(𝑥)=0 if and only if 𝑥=0_𝑋; (ii)𝜌(−𝑥)=𝜌(𝑥); (iii)𝜌(𝛼𝑥+𝛽𝑦)≤𝜌(𝑥)+𝜌(𝑦) whenever 𝛼,𝛽≥0 and 𝛼+𝛽=1.

If 𝜌 is a modular in 𝑋, then the set 𝑋_𝜌∶=𝑥∈𝑋∶𝜌(𝜆𝑥)⟶0as𝜆⟶0, (1) called a modular space, is a vector space.

As a classical example of modulars, we may give the Orlicz modular defined for every measurable real function 𝑓 by 𝜌_𝜑𝑓=_ℝ𝜑𝑓(𝑡)𝑑𝜆(𝑡), (2) where 𝜆 is the Lebesgue measure in ℝ and 𝜑∶ℝ→[0,∞) is a function satisfying some conditions. The modular space induced by the Orlicz modular 𝜌_𝜑 is called the Orlicz space. For more examples of modular spaces, we refer the reader to [2–4].

Definition 1. Let 𝑋_𝜌 be a modular space.(1)The sequence {𝑥_𝑛}⊂𝑋_𝜌 is said to be 𝜌-convergent to 𝑥∈𝑋_𝜌 if 𝜌(𝑥_𝑛−𝑥)→0, as 𝑛→∞. (2)The sequence {𝑥_𝑛}⊂𝑋_𝜌 is said to be 𝜌-Cauchy if 𝜌(𝑥_𝑛−𝑥_𝑚)→0, as 𝑛,𝑚→∞. (3)A subset 𝐶 of 𝑋_𝜌 is called 𝜌-closed if the 𝜌-limit of a 𝜌-convergent sequence of 𝐶 always belongs to 𝐶. (4)A subset 𝐶 of 𝑋_𝜌 is called 𝜌-complete if any 𝜌-Cauchy sequence in 𝐶 is 𝜌-convergent and its 𝜌-limit belongs to 𝐶.

Definition 2. The modular 𝜌 has the Fatou property if 𝜌(𝑥)≤liminf_𝑛→∞𝜌(𝑥_𝑛) whenever {𝑥_𝑛}𝜌-converges to 𝑥.

Recently, the existence and uniqueness of best proximity points in metric spaces were investigated by many authors; see [2, 5–14] and references therein. In this paper, we introduce the family of proximal quasicontraction nonself-mappings on modular spaces with the Fatou property. Our main result is a best proximity point theorem providing sufficient conditions assuring the existence and uniqueness of best proximity points for such mappings.

Let (𝐴,𝐵) be a pair of nonempty closed subsets of a modular space 𝑋_𝜌. Through this paper, we will use the following notations: 𝛾()𝜌∶,𝐴𝐴,𝐵∶=inf𝑥−𝑦𝑥,𝑦∈𝐴×𝐵₀()(),𝐵∶=𝑎∈𝐴∶𝜌𝑎−𝑏=𝛾𝐴,𝐵forsome𝑏∈𝐵₀∶=𝑏∈𝐵∶𝜌(𝑎−𝑏)=𝛾(𝐴,𝐵)forsome𝑎∈𝐴. (3)

Definition 3. Let 𝑇∶𝐴→𝐵 be a given nonself-mapping. We say that 𝑧∈𝐴₀ is a best proximity point of 𝑇 if 𝜌(𝑧−𝑇𝑧)=𝛾(𝐴,𝐵). (4)

Clearly, from condition (i), if 𝐴=𝐵, a best proximity point of 𝑇 will be a fixed point of 𝑇.

Definition 4. A nonself-mapping 𝑇∶𝐴→𝐵 is said to be a proximal quasicontraction if there exists a number 𝑞∈(0,1) such that 𝜌()()𝜌()()𝜌(),𝜌(),𝜌,𝑢−𝑇𝑥=𝛾𝐴,𝐵𝑣−𝑇𝑦=𝛾𝐴,𝐵⟹𝜌𝑢−𝑣≤𝑞max𝑥−𝑦,𝜌𝑥−𝑢𝑦−𝑣,𝜌𝑥−𝑣𝑦−𝑢 (5) where 𝑥,𝑦,𝑢,𝑣∈𝐴.

Lemma 5. Let 𝑇∶𝐴→𝐵 be a nonself-mapping. Suppose that (i)𝐴₀≠∅; (ii)𝑇(𝐴₀)⊆𝐵₀. Then, for any 𝑎∈𝐴₀, there exists a sequence {𝑥_𝑛}⊂𝐴₀ such that 𝑥₀𝜌𝑥=𝑎,_𝑛+1−𝑇𝑥_𝑛=𝛾(𝐴,𝐵),∀𝑛∈ℕ. (6)

Proof. Let 𝑎∈𝐴₀. From (ii), we have 𝑇𝑎∈𝐵₀. By definition of the set 𝐵₀, there exists 𝑥₁∈𝐴₀ such that 𝜌(𝑥₁−𝑇𝑎)=𝛾(𝐴,𝐵). Again, we have 𝑇𝑥₁∈𝐵₀, which implies that there exists 𝑥₂∈𝐴₀ such that 𝜌(𝑥₂−𝑇𝑥₁)=𝛾(𝐴,𝐵). Continuing this process, by induction, we obtain a sequence {𝑥_𝑛}⊂𝐴₀ satisfying (6).

Definition 6. Under the assumptions of Lemma 5, any sequence {𝑥_𝑛}⊂𝐴₀ satisfying (6) is called a proximal Picard sequence associated to 𝑎∈𝐴₀. We denote by 𝑃𝑃(𝑎) the set of all proximal sequences associated to 𝑎∈𝐴₀.

Definition 7. Under the assumptions of Lemma 5, we say that 𝐴₀ is proximal 𝑇-orbitally 𝜌-complete if every 𝜌-Cauchy sequence {𝑥_𝑛}∈𝑃𝑃(𝑎) for some 𝑎∈𝐴₀𝜌-converges to an element in 𝐴₀.

Let 𝑎∈𝐴₀ and {𝑥_𝑛}∈𝑃𝑃(𝑎). For all 𝑛∈ℕ, We denote 𝛿_𝑝𝑥_𝑛∶=sup𝜌𝑥_𝑛+𝑠−𝑥_𝑛+𝑟∶𝑟,𝑠∈ℕ. (7) Since 𝑥₀=𝑎, we have 𝛿_𝑝(𝑎)=sup𝜌𝑥_𝑠−𝑥_𝑟∶𝑟,𝑠∈ℕ. (8)

2. A Best Proximity Point Theorem

The following lemmas will be useful later.

Lemma 8. Let 𝑋_𝜌 be a modular space. Suppose that a nonself-mapping 𝑇∶𝐴→𝐵, where (𝐴,𝐵) is a pair of subsets of 𝑋, satisfies the following conditions:(i)∃𝑎∈𝐴₀|𝛿_𝑝(𝑎)<∞; (ii)𝑇(𝐴₀)⊆𝐵₀; (iii)𝑇 is proximal quasi-contraction. Then, for any {𝑥_𝑛}∈𝑃𝑃(𝑎), one has 𝜌𝑥_𝑛−𝑥_𝑛+𝑚≤𝛿_𝑝𝑥_𝑛≤𝑞^𝑛𝛿_𝑝(𝑎), (9) for any 𝑛≥1 and 𝑚∈ℕ.

Proof. Let {𝑥_𝑛}∈𝑃𝑃(𝑎) and (𝑠,𝑟)∈ℕ². From the definition of 𝑃𝑃(𝑎), for all 𝑛≥1, we have 𝜌𝑥_𝑛+𝑠−𝑇𝑥_{𝑛−1+𝑠}=𝜌𝑥_𝑛+𝑟−𝑇𝑥_{𝑛−1+𝑟}=𝛾(𝐴,𝐵), (10) which implies, since 𝑇 is a proximal quasi-contraction, that 𝜌𝑥_𝑛+𝑠−𝑥_𝑛+𝑟≤𝑞max𝜌𝑥_{𝑛−1+𝑠}−𝑥_{𝑛−1+𝑟},𝜌𝑥_{𝑛−1+𝑠}−𝑥_𝑛+𝑠,𝜌𝑥_{𝑛−1+𝑟}−𝑥_𝑛+𝑟,𝜌𝑥_{𝑛−1+𝑠}−𝑥_𝑛+𝑟,𝜌𝑥_{𝑛−1+𝑟}−𝑥_𝑛+𝑠≤𝑞𝛿_𝑝𝑥_𝑛−1. (11) This implies immediately that 𝛿_𝑝𝑥_𝑛≤𝑞𝛿_𝑝𝑥_𝑛−1, (12) for all 𝑛≥1. Hence, for any 𝑛∈ℕ, we have 𝛿_𝑝𝑥_𝑛≤𝑞^𝑛𝛿_𝑝(𝑎). (13) Using the above inequality, for all 𝑛≥1 and 𝑚∈ℕ, we have 𝜌𝑥_𝑛−𝑥_𝑛+𝑚≤𝛿_𝑝𝑥_𝑛≤𝑞^𝑛𝛿_𝑝(𝑎). (14)

Lemma 9. Let (𝐴,𝐵) be a pair of subsets of a modular space 𝑋_𝜌. Let 𝑇∶𝐴→𝐵 be a given nonself-mapping. Suppose that (i)𝐴₀ is proximal 𝑇-orbitally 𝜌-complete; (ii)𝑇(𝐴₀)⊆𝐵₀; (iii)∃𝑎∈𝐴₀ such that 𝛿_𝑝(𝑎)<∞; (iv)𝑇 is proximal quasi-contraction; (v)𝜌 satisfies the Fatou property. Then, any sequence {𝑥_𝑛}∈𝑃𝑃(𝑎)𝜌-converges to some 𝑧∈𝐴₀ such that 𝜌𝑥_𝑛−𝑧≤𝑞^𝑛𝛿_𝑝(𝑎), (15) for all 𝑛≥1. Moreover, there exists 𝑤∈𝐴₀ such that 𝜌(𝑤−𝑇𝑧)=𝛾(𝐴,𝐵). (16)

Proof. Let {𝑥_𝑛}∈𝑃𝑃(𝑎). From Lemma 8, we know that {𝑥_𝑛} is 𝜌-Cauchy. Since 𝐴₀ is proximal 𝑇-orbitally 𝜌-complete, then there exists 𝑧∈𝐴₀ such that {𝑥_𝑛}𝜌-converges to 𝑧. Again, by Lemma 8, we have 𝜌𝑥_𝑛−𝑥_𝑛+𝑚≤𝑞^𝑛𝛿_𝑝(𝑎), (17) for any 𝑛≥1 and 𝑚∈ℕ. Letting 𝑚→∞ in the above inequality and using the Fatou property, we obtain 𝜌𝑥_𝑛−𝑧≤𝑞^𝑛𝛿_𝑝(𝑎), (18) for all 𝑛≥1. Now, since 𝑇𝑧∈𝐵₀, by the definition of 𝐵₀, there exists some 𝑤∈𝐴₀ such that 𝜌(𝑤−𝑇𝑧)=𝛾(𝐴,𝐵).

Now, we are ready to state and prove our main result.

Theorem 10. Suppose that the assumptions of the previous lemma are satisfied. Assume 𝜌(𝑧−𝑤)<∞ and 𝜌(𝑎−𝑤)<∞. Then, the 𝜌-limit 𝑧∈𝐴₀ of {𝑥_𝑛}∈𝑃𝑃(𝑎) is a best proximity point of 𝑇. Moreover, if 𝑢∈𝐴₀ is any best proximity point of 𝑇 such that 𝜌(𝑧−𝑢)<∞, then one has 𝑧=𝑢.

Proof. By Lemma 9, we have 𝜌(𝑤−𝑇𝑧)=𝛾(𝐴,𝐵). (19) On the other hand, from the definition of {𝑥_𝑛}, we have 𝜌𝑥₁−𝑇𝑎=𝛾(𝐴,𝐵). (20) Since 𝑇 is proximal quasi-contraction, we get that 𝜌𝑤−𝑥₁𝜌()(),𝜌≤𝑞max𝑎−𝑧,𝜌𝑧−𝑤𝑎−𝑥₁,𝜌𝑧−𝑥₁,𝜌(𝑎−𝑤). (21) Using Lemmas 8 and 9, we obtain that 𝜌𝑤−𝑥₁≤max𝑞𝛿_𝑝(𝑎),𝑞𝜌(𝑧−𝑤),𝑞𝜌(𝑎−𝑤). (22) Again, from the definition of {𝑥_𝑛}, we have 𝜌𝑥₂−𝑇𝑥₁=𝛾(𝐴,𝐵). (23) Since 𝑇 is proximal quasi-contraction, we get that 𝜌𝑤−𝑥₂≤𝑞max𝜌𝑧−𝑥₁,𝜌(𝑧−𝑤),𝜌𝑥₁−𝑥₂,𝜌𝑧−𝑥₂,𝜌𝑥₁−𝑤≤𝑞max𝑞𝛿_𝑝(𝑎),𝜌(𝑧−𝑤),𝛿_𝑝𝑥₁,𝑞²𝛿_𝑝(𝑎),𝜌𝑥₁−𝑤≤𝑞max𝑞𝛿_𝑝(𝑎),𝜌(𝑧−𝑤),𝑞𝛿_𝑝(𝑎),𝑞²𝛿_𝑝(𝑎),𝜌𝑥₁−𝑤=𝑞max𝑞𝛿_𝑝(𝑎),𝜌(𝑧−𝑤),𝜌𝑥₁((𝑞−𝑤from22))≤max²𝛿_𝑝(𝑎),𝑞𝜌(𝑧−𝑤),𝑞²𝜌(𝑎−𝑤). (24) Thus, we proved that 𝜌𝑤−𝑥₂≤max𝑞²𝛿_𝑝(𝑎),𝑞𝜌(𝑧−𝑤),𝑞²𝜌(𝑎−𝑤). (25) Continuing this process, by induction, we get that 𝜌𝑤−𝑥_𝑛≤max𝑞^𝑛𝛿_𝑝(𝑎),𝑞𝜌(𝑧−𝑤),𝑞^𝑛𝜌(𝑎−𝑤), (26) for all 𝑛≥1. Therefore, we have limsup_𝑛→∞𝜌𝑥_𝑛−𝑤≤𝑞𝜌(𝑧−𝑤). (27) Using the Fatou property, we get 𝜌(𝑧−𝑤)≤𝑞𝜌(𝑧−𝑤), (28) which implies, since 𝑞<1, that 𝜌(𝑧−𝑤)=0; that is, 𝑧=𝑤. Thus, from (19), we get that 𝜌(𝑧−𝑇𝑧)=𝛾(𝐴,𝐵). (29) Hence, 𝑧 is a best proximity point of 𝑇.
Suppose now that 𝑢∈𝐴₀ is a best proximity point of 𝑇 such that 𝜌(𝑧−𝑢)<∞. Since 𝑇 is proximal quasi-contraction, we obtain that 𝜌()𝜌()()(),𝜌()(().𝑧−𝑢≤𝑞max𝑧−𝑢,𝜌𝑧−𝑧,𝜌𝑢−𝑢𝑧−𝑢,𝜌𝑢−𝑧)=𝑞𝜌𝑧−𝑢 (30) Since 𝑞<1, we have 𝜌(𝑧−𝑢)=0, which implies that 𝑢=𝑧.

Consider now the case 𝐴=𝐵. In this case, a best proximity point of 𝑇∶𝐴→𝐵 will be a fixed point of the self-mapping 𝑇.

Definition 11. We say that 𝐴 is 𝑇-orbitally 𝜌-complete if {𝑇^𝑛𝑎} is a 𝜌-Cauchy for every 𝑎∈𝐴, then it is 𝜌-convergent to an element of 𝐴.

Similarly to Ćirić [15] definition, Khamsi [16] introduced the concept of quasicontraction self-mappings in modular spaces.

Definition 12. The self-mapping 𝑇∶𝐴→𝐴 is said to be a quasicontraction if there exists a constant 𝑞∈(0,1) such that 𝜌𝜌(),𝜌,𝑇𝑥−𝑇𝑦≤𝑞max𝑥−𝑦,𝜌𝑥−𝑇𝑥,𝜌𝑦−𝑇𝑦𝑥−𝑇𝑦,𝜌𝑦−𝑇𝑥 (31) for all 𝑥,𝑦∈𝐴.

From Theorem 10, we can deduce the following result, that is, a slight extension of the fixed point theorem established by Khamsi in [16].

Corollary 13. Consider a self-mapping 𝑇∶𝐴→𝐴, where 𝐴 is a nonempty subset of 𝑋_𝜌. Suppose that the following conditions hold:(i)𝐴 is 𝑇-orbitally 𝜌-complete; (ii)∃𝑎∈𝐴 such that sup{𝜌(𝑇^𝑠𝑎−𝑇^𝑟𝑎)∶𝑠,𝑟∈ℕ}<∞; (iii)𝜌 satisfies the Fatou property; (iv)𝑇 is quasi-contraction. Then, the sequence {𝑇^𝑛𝑎}𝜌-converges to some 𝑧∈𝐴. Moreover, if 𝜌(𝑧−𝑇𝑧)<∞ and 𝜌(𝑎−𝑇𝑧)<∞, then 𝑧 is a fixed point of  𝑇. If 𝑢∈𝐴 is a fixed point of  𝑇 with 𝜌(𝑧−𝑢)<∞, then 𝑢=𝑧.

Acknowledgment

This work is supported by the Research Center, College of Science, King Saud University.